Non-oscillatory central schemes for one- and two-dimensional MHD equations: I

Abstract

The computations reported in this paper demonstrate the remarkable versatility of central schemes as black-box, Jacobian-free solvers for ideal magnetohydrodynamics (MHD) equations. Here we utilize a family of high-resolution, non-oscillatory central schemes for the approximate solution of the one- and two-dimensional MHD equations. We present simulations based on staggered grids of several MHD prototype problems. Solution of one-dimensional shock-tube problems is carried out using second- and third-order central schemes [Numer. Math. 79 (1998) 397; J. Comput. Phys. 87 (2) (1990) 408], and we use the second-order central scheme [SIAM J. Sci Comput. 19 (6) (1998) 1892] which is adapted for the solution of the two-dimensional Kelvin–Helmholtz and Orszag–Tang problems. A qualitative comparison reveals an excellent agreement with previous results based on upwind schemes. Central schemes, however, require little knowledge about the eigenstructure of the problem – in fact, we even avoid an explicit evaluation of the corresponding Jacobians, while at the same time they eliminate the need for dimensional splitting.